arXiv subjects

Kevin Tian

Publications and source records attributed to Kevin Tian.

59 records · Page 4Linked to original sources

arXiv2019

A Rank-1 Sketch for Matrix Multiplicative Weights

We show that a simple randomized sketch of the matrix multiplicative weight (MMW) update enjoys (in expectation) the same regret bounds as MMW, up to a small constant factor. Unlike MMW, where every step requires full matrix exponentiation, our steps require only a single product of the form $e^A b$, which the Lanczos method approximates efficiently. Our key technique is to view the sketch as a $\textit{randomized mirror projection}$, and perform mirror descent analysis on the $\textit{expected projection}$. Our sketch solves the online eigenvector problem, improving the best known complexity bounds by $Ω(\log^5 n)$. We also apply this sketch to semidefinite programming in saddle-point form, yielding a simple primal-dual scheme with guarantees matching the best in the literature.

cs.LG↗

arXiv2019

A Direct $\tilde{O}(1/ε)$ Iteration Parallel Algorithm for Optimal Transport

Optimal transportation, or computing the Wasserstein or ``earth mover's'' distance between two distributions, is a fundamental primitive which arises in many learning and statistical settings. We give an algorithm which solves this problem to additive $ε$ with $\tilde{O}(1/ε)$ parallel depth, and $\tilde{O}\left(n^2/ε\right)$ work. Barring a breakthrough on a long-standing algorithmic open problem, this is optimal for first-order methods. Blanchet et. al. '18, Quanrud '19 obtained similar runtimes through reductions to positive linear programming and matrix scaling. However, these reduction-based algorithms use complicated subroutines which may be deemed impractical due to requiring solvers for second-order iterations (matrix scaling) or non-parallelizability (positive LP). The fastest practical algorithms run in time $\tilde{O}(\min(n^2 / ε^2, n^{2.5} / ε))$ (Dvurechensky et. al. '18, Lin et. al. '19). We bridge this gap by providing a parallel, first-order, $\tilde{O}(1/ε)$ iteration algorithm without worse dependence on dimension, and provide preliminary experimental evidence that our algorithm may enjoy improved practical performance. We obtain this runtime via a primal-dual extragradient method, motivated by recent theoretical improvements to maximum flow (Sherman '17).

cs.DS↗

arXiv2018

CoVeR: Learning Covariate-Specific Vector Representations with Tensor Decompositions

Word embedding is a useful approach to capture co-occurrence structures in large text corpora. However, in addition to the text data itself, we often have additional covariates associated with individual corpus documents---e.g. the demographic of the author, time and venue of publication---and we would like the embedding to naturally capture this information. We propose CoVeR, a new tensor decomposition model for vector embeddings with covariates. CoVeR jointly learns a \emph{base} embedding for all the words as well as a weighted diagonal matrix to model how each covariate affects the base embedding. To obtain author or venue-specific embedding, for example, we can then simply multiply the base embedding by the associated transformation matrix. The main advantages of our approach are data efficiency and interpretability of the covariate transformation. Our experiments demonstrate that our joint model learns substantially better covariate-specific embeddings compared to the standard approach of learning a separate embedding for each covariate using only the relevant subset of data, as well as other related methods. Furthermore, CoVeR encourages the embeddings to be "topic-aligned" in that the dimensions have specific independent meanings. This allows our covariate-specific embeddings to be compared by topic, enabling downstream differential analysis. We empirically evaluate the benefits of our algorithm on datasets, and demonstrate how it can be used to address many natural questions about covariate effects. Accompanying code to this paper can be found at http://github.com/kjtian/CoVeR.

cs.CL↗

arXiv2017

Learning Populations of Parameters

Consider the following estimation problem: there are $n$ entities, each with an unknown parameter $p_i \in [0,1]$, and we observe $n$ independent random variables, $X_1,\ldots,X_n$, with $X_i \sim $ Binomial$(t, p_i)$. How accurately can one recover the "histogram" (i.e. cumulative density function) of the $p_i$'s? While the empirical estimates would recover the histogram to earth mover distance $Θ(\frac{1}{\sqrt{t}})$ (equivalently, $\ell_1$ distance between the CDFs), we show that, provided $n$ is sufficiently large, we can achieve error $O(\frac{1}{t})$ which is information theoretically optimal. We also extend our results to the multi-dimensional parameter case, capturing settings where each member of the population has multiple associated parameters. Beyond the theoretical results, we demonstrate that the recovery algorithm performs well in practice on a variety of datasets, providing illuminating insights into several domains, including politics, sports analytics, and variation in the gender ratio of offspring.

cs.LG↗

arXiv2014

Variance on the Leaves of a Tree Markov Random Field: Detecting Character Dependencies in Phylogenies

Stochastic models of evolution (Markov random fields on trivalent trees) generally assume that different characters (different runs of the stochastic process) are independent and identically distributed. In this paper we take the first steps towards addressing dependent characters. Specifically we show that, under certain technical assumptions regarding the evolution of individual characters, we can detect any significant, history independent, correlation between any pair of multistate characters. For the special case of the Cavender-Farris-Neyman (CFN) model on two states with symmetric transition matrices, our analysis needs milder assumptions. To perform the analysis, we need to prove a new concentration result for multistate random variables of a Markov random field on arbitrary trivalent trees: we show that the random variable counting the number of leaves in any particular subset of states has variance that is subquadratic in the number of leaves.

q-bio.PE↗